Purchase Solution

8 Topology Questions : Hausdorff Space, Countability, Compactness and Homomorphisms

Not what you're looking for?

Ask Custom Question

1. Show that the collection
13 {[O,c)J 0<c 1}u{(d,1]|O<d< 1)
cannot be a base for the subspace topology [O, 1], where is the Euclidean topology on rt
Hint: Use contradiction, i.e., first assume that B is a base for [O, 1].
2. Let B {[a, b) x [c, d)Ia < b, : < d}. Show that B is a base for the product space (R x R, £ x £), where £ is the lower limit topology on 1?.
3. Let X. Y be spaces and f : X ?> Y a continuous surjcction. Prove that if X is P countable, so is Y.
4. Let X, Y he spaces and f : X ?> Y a continuous surjection. Prove that if X is compact, so is V.
3. Show that the union of finitely many compact subsets of a space is compact.
6. Let X be a compact space and V a Hausdorff space. Then show that f X ?> Y is a homomorphism if and only if f is bijeetive and continuous.
7. Let X be a Hausdorff space and P, Q disjoint compact subsets of X. Then show that there exist open sets U and V such that P C U, Q C V and U U V 0.
This property tells us that compact subsets of a Hausdorif space behaves like points.
8. Let X, Y be spaces arid f X ?> V a continuous surjection. Prove that if X is connected, so is V.

Attachments
Purchase this Solution

Solution Summary

Eight topology questions involving Hausdorff Space, Countability, Compactness and Homomorphisms. The solution is detailed and well presented.

Purchase this Solution


Free BrainMass Quizzes
Probability Quiz

Some questions on probability

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.